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Math Help - Parametric equations for cylinder/plane intersection

  1. #1
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    Parametric equations for cylinder/plane intersection

    I'd like to make sure my thinking is right on this problem and to see what the next step would be because I'm getting stuck.

    The problem is to find the parametric equations for the ellipse which made by the intersection of a right circular cylinder of radius c with the plane which intersects the z-axis at point 'a' and the y-axis at point 'b' when t=0.

    I think the equation for the cylinder would be x^2+y^2=c^2

    As for the plane I am less sure about the equation. I have points and I could write an equation with one if I had a normal vector to the plane... or the cross product of two vectors in it. Two vectors would be between the points (0, b, 0) and (0, 0, a) and the points (0, b, 0) and (1, 0, a) since x will take on all values. Those vectors are <0, -b, a> and <1, -b, a>

    \left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 0 & -b & a \\ 1 & -b & a\end{matrix}\right| Which yields  0\mathbf{i}  + a\mathbf{j} + b\mathbf{k}

    so using the y intercept and n_1(x-x_1) + n_2(y-y_1) + n_3(z-z_1) = 0 the equation of the plane should be  a(y - b) + bz = 0 or ay - ab +bz = 0

    I get stuck here though... what should I do next? Any help is appreciated.

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    Quote Originally Posted by Shananay View Post
    I'd like to make sure my thinking is right on this problem and to see what the next step would be because I'm getting stuck.

    The problem is to find the parametric equations for the ellipse which made by the intersection of a right circular cylinder of radius c with the plane which intersects the z-axis at point 'a' and the y-axis at point 'b' when t=0.

    I think the equation for the cylinder would be x^2+y^2=c^2

    As for the plane I am less sure about the equation. I have points and I could write an equation with one if I had a normal vector to the plane... or the cross product of two vectors in it. Two vectors would be between the points (0, b, 0) and (0, 0, a) and the points (0, b, 0) and (1, 0, a) since x will take on all values. Those vectors are <0, -b, a> and <1, -b, a>

    \left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 0 & -b & a \\ 1 & -b & a\end{matrix}\right| Which yields  0\mathbf{i}  + a\mathbf{j} + b\mathbf{k}

    so using the y intercept and n_1(x-x_1) + n_2(y-y_1) + n_3(z-z_1) = 0 the equation of the plane should be  a(y - b) + bz = 0 or ay - ab +bz = 0

    I get stuck here though... what should I do next? Any help is appreciated.

    With the given conditions you'll get a family of planes with the line

    l: (x,y,z)=(0,0,a)+s(0,-b,a)

    All planes of the family have this line in common. (Line of intersection of all planes)

    In general: A plane is determined by 3 points. So I assume that there are additional conditions which you have to use.
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    Quote Originally Posted by earboth View Post
    With the given conditions you'll get a family of planes with the line

    l: (x,y,z)=(0,0,a)+s(0,-b,a)

    All planes of the family have this line in common. (Line of intersection of all planes)

    In general: A plane is determined by 3 points. So I assume that there are additional conditions which you have to use.
    Is it enough to say that the plane never intersects the x-axis? I drew the problem from the board and thought I copied all the information. Short of giving another point of the plane explicitly what kind of additional information could there be?
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    So I just want to be sure, this problem is not solvable with the given information because I cannot identify a unique plane?
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    Quote Originally Posted by Shananay View Post
    Is it enough to say that the plane never intersects the x-axis? I drew the problem from the board and thought I copied all the information. Short of giving another point of the plane explicitly what kind of additional information could there be?
    If the plane never intersects the x-axis then you have actually the necessary 3rd point at P(c, 0, a). This point P belongs to the ellipse which is produced by the plane when intersecting the surface of the cylinder.

    The plane is determined by the vectors:

    \overrightarrow{b - a} = (0, b, -a) and \vec v = (c, 0, 0)

    The plane has to pass through (0, 0, a) and the normal vector of the plane must be:

    (0,b,-a) \times (c,0,0) = (0, ca, bc) = c \cdot (0,a,b)

    Therefore the equation of the plane is:

    (0,a,b)((x,y,z) - (0,0,a))=0~\implies~ay+bz-ab=0
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    Quote Originally Posted by earboth View Post
    Therefore the equation of the plane is:

    (0,a,b)((x,y,z) - (0,0,a))=0~\implies~ay+bz-ab=0
    Okay, this is what I had for the plane. But I am looking for the equations for the ellipse. I don't know how to incorporate the t=0 point and I don't see what the parametrization would look like.
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    Quote Originally Posted by Shananay View Post
    Okay, this is what I had for the plane. But I am looking for the equations for the ellipse. I don't know how to incorporate the t=0 point and I don't see what the parametrization would look like.
    1. The condition t = 0 has nothing to do with the ellipse.

    2. You already have a parametric equation of the ellipse:

    e:\left\{\begin{array}{l}x=s \\ y = \sqrt{c^2-s^2} \\ z=a-\frac ab\cdot \sqrt{c^2-s^2} \end{array} \right.

    where s is a real variable with -c\leq s\leq c
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    Quote Originally Posted by Shananay View Post
    Okay, this is what I had for the plane. But I am looking for the equations for the ellipse. I don't know how to incorporate the t=0 point and I don't see what the parametrization would look like.
    To demonstrate the effects of my previous calculations I've attached a drawing showing the cylinder with the ellipse and the cylinder with the corresponding plane.
    Attached Thumbnails Attached Thumbnails Parametric equations for cylinder/plane intersection-zylschnitt_ebene.png  
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    Quote Originally Posted by earboth View Post
    2. You already have a parametric equation of the ellipse:

    e:\left\{\begin{array}{l}x=s \\ y = \sqrt{c^2-s^2} \\ z=a-\frac ab\cdot \sqrt{c^2-s^2} \end{array} \right.

    where s is a real variable with -c\leq s\leq c
    I did not, but I do now. I'm not having trouble visualizing the picture, I just don't know how to get those parametric equations. If I already had them I wouldn't have been asking the question.
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    Re: Parametric equations for cylinder/plane intersection

    Quote Originally Posted by earboth View Post
    1. The condition t = 0 has nothing to do with the ellipse.

    2. You already have a parametric equation of the ellipse:

    e:\left\{\begin{array}{l}x=s \\ y = \sqrt{c^2-s^2} \\ z=a-\frac ab\cdot \sqrt{c^2-s^2} \end{array} \right.

    where s is a real variable with -c\leq s\leq c
    I realize this is a couple years post-post, but I should probably point out that the following expression needs to have a plus/minus in both the Y and Z parameters:

    √(c^2-s^2)

    When the Y parameter is positive, the Z parameter should use the minus.

    Great work getting the equations, though. It was very helpful to me.
    Last edited by bmullins; July 5th 2012 at 11:30 AM.
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